Answer:
Explanation:
The system of inequalities is:
1. 2x - 3y ≥ 10
2. 3x + 4y ≤ 5
To find the ordered pairs that are solutions to this system of inequalities, we need to graph the two inequalities on a coordinate plane and identify the region that satisfies both of them.
For the first inequality, 2x - 3y ≥ 10, we can rewrite it as:
-3y ≥ -2x + 10
y ≤ (2/3)x - (10/3)
For the second inequality, 3x + 4y ≤ 5, we can rewrite it as:
4y ≤ -3x + 5
y ≤ (-3/4)x + (5/4)
Now, let's graph these two lines on a coordinate plane:
1. Graph the line y = (2/3)x - (10/3):
- Plot the y-intercept at (0, -10/3).
- Use the slope (rise over run) of 2/3 to find another point on the line. For example, if we move 3 units to the right, we go up 2 units, giving us the point (3, -8/3).
- Draw a solid line through these two points to represent the inequality y ≤ (2/3)x - (10/3).
2. Graph the line y = (-3/4)x + (5/4):
- Plot the y-intercept at (0, 5/4).
- Use the slope (rise over run) of -3/4 to find another point on the line. For example, if we move 4 units to the right, we go down 3 units, giving us the point (4, 2/4).
- Draw a solid line through these two points to represent the inequality y ≤ (-3/4)x + (5/4).
The shaded region where both inequalities are satisfied is the solution to the system. It is the region below or on the line y = (2/3)x - (10/3), and below or on the line y = (-3/4)x + (5/4).
The ordered pairs that are solutions to this system of inequalities are all the points in this shaded region. To find specific ordered pairs, you can choose any point within this region and list its x and y coordinates.
For example, one possible solution is (-1, -2), because it satisfies both inequalities:
1. 2(-1) - 3(-2) = 2 + 6 = 8 ≥ 10 (satisfied)
2. 3(-1) + 4(-2) = -3 - 8 = -11 ≤ 5 (satisfied)